MAYBE

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict Trs:
  { f(X) -> cons(X, n__f(n__g(X)))
  , f(X) -> n__f(X)
  , g(X) -> n__g(X)
  , g(0()) -> s(0())
  , g(s(X)) -> s(s(g(X)))
  , sel(0(), cons(X, Y)) -> X
  , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
  , activate(X) -> X
  , activate(n__f(X)) -> f(activate(X))
  , activate(n__g(X)) -> g(activate(X)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We add following dependency tuples:

Strict DPs:
  { f^#(X) -> c_1()
  , f^#(X) -> c_2()
  , g^#(X) -> c_3()
  , g^#(0()) -> c_4()
  , g^#(s(X)) -> c_5(g^#(X))
  , sel^#(0(), cons(X, Y)) -> c_6()
  , sel^#(s(X), cons(Y, Z)) ->
    c_7(sel^#(X, activate(Z)), activate^#(Z))
  , activate^#(X) -> c_8()
  , activate^#(n__f(X)) -> c_9(f^#(activate(X)), activate^#(X))
  , activate^#(n__g(X)) -> c_10(g^#(activate(X)), activate^#(X)) }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { f^#(X) -> c_1()
  , f^#(X) -> c_2()
  , g^#(X) -> c_3()
  , g^#(0()) -> c_4()
  , g^#(s(X)) -> c_5(g^#(X))
  , sel^#(0(), cons(X, Y)) -> c_6()
  , sel^#(s(X), cons(Y, Z)) ->
    c_7(sel^#(X, activate(Z)), activate^#(Z))
  , activate^#(X) -> c_8()
  , activate^#(n__f(X)) -> c_9(f^#(activate(X)), activate^#(X))
  , activate^#(n__g(X)) -> c_10(g^#(activate(X)), activate^#(X)) }
Weak Trs:
  { f(X) -> cons(X, n__f(n__g(X)))
  , f(X) -> n__f(X)
  , g(X) -> n__g(X)
  , g(0()) -> s(0())
  , g(s(X)) -> s(s(g(X)))
  , sel(0(), cons(X, Y)) -> X
  , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
  , activate(X) -> X
  , activate(n__f(X)) -> f(activate(X))
  , activate(n__g(X)) -> g(activate(X)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We estimate the number of application of {1,2,3,4,6,8} by
applications of Pre({1,2,3,4,6,8}) = {5,7,9,10}. Here rules are
labeled as follows:

  DPs:
    { 1: f^#(X) -> c_1()
    , 2: f^#(X) -> c_2()
    , 3: g^#(X) -> c_3()
    , 4: g^#(0()) -> c_4()
    , 5: g^#(s(X)) -> c_5(g^#(X))
    , 6: sel^#(0(), cons(X, Y)) -> c_6()
    , 7: sel^#(s(X), cons(Y, Z)) ->
         c_7(sel^#(X, activate(Z)), activate^#(Z))
    , 8: activate^#(X) -> c_8()
    , 9: activate^#(n__f(X)) -> c_9(f^#(activate(X)), activate^#(X))
    , 10: activate^#(n__g(X)) ->
          c_10(g^#(activate(X)), activate^#(X)) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { g^#(s(X)) -> c_5(g^#(X))
  , sel^#(s(X), cons(Y, Z)) ->
    c_7(sel^#(X, activate(Z)), activate^#(Z))
  , activate^#(n__f(X)) -> c_9(f^#(activate(X)), activate^#(X))
  , activate^#(n__g(X)) -> c_10(g^#(activate(X)), activate^#(X)) }
Weak DPs:
  { f^#(X) -> c_1()
  , f^#(X) -> c_2()
  , g^#(X) -> c_3()
  , g^#(0()) -> c_4()
  , sel^#(0(), cons(X, Y)) -> c_6()
  , activate^#(X) -> c_8() }
Weak Trs:
  { f(X) -> cons(X, n__f(n__g(X)))
  , f(X) -> n__f(X)
  , g(X) -> n__g(X)
  , g(0()) -> s(0())
  , g(s(X)) -> s(s(g(X)))
  , sel(0(), cons(X, Y)) -> X
  , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
  , activate(X) -> X
  , activate(n__f(X)) -> f(activate(X))
  , activate(n__g(X)) -> g(activate(X)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ f^#(X) -> c_1()
, f^#(X) -> c_2()
, g^#(X) -> c_3()
, g^#(0()) -> c_4()
, sel^#(0(), cons(X, Y)) -> c_6()
, activate^#(X) -> c_8() }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { g^#(s(X)) -> c_5(g^#(X))
  , sel^#(s(X), cons(Y, Z)) ->
    c_7(sel^#(X, activate(Z)), activate^#(Z))
  , activate^#(n__f(X)) -> c_9(f^#(activate(X)), activate^#(X))
  , activate^#(n__g(X)) -> c_10(g^#(activate(X)), activate^#(X)) }
Weak Trs:
  { f(X) -> cons(X, n__f(n__g(X)))
  , f(X) -> n__f(X)
  , g(X) -> n__g(X)
  , g(0()) -> s(0())
  , g(s(X)) -> s(s(g(X)))
  , sel(0(), cons(X, Y)) -> X
  , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
  , activate(X) -> X
  , activate(n__f(X)) -> f(activate(X))
  , activate(n__g(X)) -> g(activate(X)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { activate^#(n__f(X)) -> c_9(f^#(activate(X)), activate^#(X)) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { g^#(s(X)) -> c_1(g^#(X))
  , sel^#(s(X), cons(Y, Z)) ->
    c_2(sel^#(X, activate(Z)), activate^#(Z))
  , activate^#(n__f(X)) -> c_3(activate^#(X))
  , activate^#(n__g(X)) -> c_4(g^#(activate(X)), activate^#(X)) }
Weak Trs:
  { f(X) -> cons(X, n__f(n__g(X)))
  , f(X) -> n__f(X)
  , g(X) -> n__g(X)
  , g(0()) -> s(0())
  , g(s(X)) -> s(s(g(X)))
  , sel(0(), cons(X, Y)) -> X
  , sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
  , activate(X) -> X
  , activate(n__f(X)) -> f(activate(X))
  , activate(n__g(X)) -> g(activate(X)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

We replace rewrite rules by usable rules:

  Weak Usable Rules:
    { f(X) -> cons(X, n__f(n__g(X)))
    , f(X) -> n__f(X)
    , g(X) -> n__g(X)
    , g(0()) -> s(0())
    , g(s(X)) -> s(s(g(X)))
    , activate(X) -> X
    , activate(n__f(X)) -> f(activate(X))
    , activate(n__g(X)) -> g(activate(X)) }

We are left with following problem, upon which TcT provides the
certificate MAYBE.

Strict DPs:
  { g^#(s(X)) -> c_1(g^#(X))
  , sel^#(s(X), cons(Y, Z)) ->
    c_2(sel^#(X, activate(Z)), activate^#(Z))
  , activate^#(n__f(X)) -> c_3(activate^#(X))
  , activate^#(n__g(X)) -> c_4(g^#(activate(X)), activate^#(X)) }
Weak Trs:
  { f(X) -> cons(X, n__f(n__g(X)))
  , f(X) -> n__f(X)
  , g(X) -> n__g(X)
  , g(0()) -> s(0())
  , g(s(X)) -> s(s(g(X)))
  , activate(X) -> X
  , activate(n__f(X)) -> f(activate(X))
  , activate(n__g(X)) -> g(activate(X)) }
Obligation:
  innermost runtime complexity
Answer:
  MAYBE

The input cannot be shown compatible

Arrrr..